Are we all log-normal deviates?
Some simple human characteristics are reasonably close to normal – like height or length of index finger. But more complex human abilities are strongly positively skew. Below is a cute little 90 second talk from Angela Duckworth, a well-known academic pyschologist at the University of Pennsylvania
Click HERE to hear the short address.
I do not know what the data actually say, but the basic argument seems sound. Complex human abilities depend on several contingent abilities that intereact. For instance, tennis requires hand-to-eye, athleticism, fitness and psychological hardness. And it sounds more reasonable that the final achievement would be a product of these abilities rather than a sum. Similarly. skills like communication and organization don’t just contribute additively as they would on a report card; they are multipliers that amplify your effectiveness in other areas.
But to get the log-normal distribution for the final performance, the basic abilities to be multiplied also have to be log-normal. Why would the basic abilities be log-normal? Because variation on the log-scale is variation on the proportional scale. It seems reasonable that you are as likely to be twice as good as average as twice as bad as average. Not to mention that most traits have to be positive.
On the other hand, IQ tests are, I believe, deliberately scaled to be normal and heights within a racial-gender group are close to normal but with heavy tails, due to genetic abnormalities.
That got me to thinking. Normals are additive. Log-normals are additive on the log scale and so are multiplicative. In between, you could also have additivity on the Box-Cox scale. This is fairly standard for regression but I have never heard a name for the 3-parameter family of distributions where
Y=(Xlambda-1)/lambda
is normal - in other words X has the distribution of (Y lambda+1)1/lambda.
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October 19th, 2009 at 1:24 pm
You might just as well say \where x^lambda is normal\ as for non-zero lambda x^lambda is normal whenever
Y=(x^lambda-1)/lambda is.
October 22nd, 2009 at 1:25 pm
I think it was Kolmorov (or Cramer?) who explained the log-normal character of sand particle size by imagining rocks splitting in two, with the proportions going either side of the split uniform (0,1). Take logs and invoke the central limit theorem, I suppose.
That human abilities do have a strong positive skewness is clear, though. My favourite example of this is the old Cambridge tripos exams. The exam itself was a close to openended as you could get, with a maximum possible mark of 1000. Typically the best candidate would score ~600, the next ~200 and the others all less than 100. It makes you wonder how much scoring mathematics exams on a strict (0,100) scale, as we do in both matriculation exams and in most University courses, really distorts the upper tail.
Bill.
October 23rd, 2009 at 9:33 pm
I can’t point to any theory but it seems to me that any physical measure bounded on one side (at zero) and unbounded on the other would tend to be skew, especially if the mean is only a few standard deviations from zero.
Nor do think it necessary to invoke a multiplicative component trait mechanism to explain this. If we propose a Markov?) process where a trait of next generation is some random percentage (<100%) greater or less than that of the previous generation, the distribution of a trait will be skew (and given suitable conditions may tend to log-normal), provided the distribution of that random percetage is symmetrical about 0%.
As a trivial example, start with 1.0 and add or subtract (at random) 50% three times. The set of all possible numbers generated will 0.125, 0.25, 0.5, 1.0, 1.5, 2.25, 3.325.
October 24th, 2009 at 8:49 pm
Just to correct that last example. The set of all possible numbers after three trials is:
0.125, 0.25, 0.375(3 times), 0.5, 0.75 (twice), 1.0, 1.125 (3 times), 1.5, 2.25 and 3.375. The mean is 1.0, but the distribution is clearly positive skew. The set of third generation numbers is 0.125, 0.375(3 times), 1.125 (3 times) and 3.375. The mean is still one. On a log scale this third generation has a skew of zero, as does the 4th. Perhaps I could conjecture the nth generation will approach log-normal in asymptote.
For reasons I don’t fully understand a simple (Markov?) chain will not generate this distribution, but will converge on zero.
January 21st, 2010 at 5:17 am
As a trivial example, start with 1.0 and add or subtract (at random) 50% three times. The set of all possible numbers generated will 0.125, 0.25, 0.5, 1.0, 1.5, 2.25, 3.325.